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Zorluk: OrtaSimplifying Expressions and Combining Like Terms

Match each of the unsimplified algebraic expressions on the left with its equivalent simplified form on the right. (Assume all variables represent real numbers.)

  • 3r(2r−s)−2s(r−3s)3r(2r - s) - 2s(r - 3s)6r2−5rs+6s26r^2 - 5rs + 6s^2
  • (2r−s)2−2(r2−rs)(2r - s)^2 - 2(r^2 - rs)2r2−2rs+s22r^2 - 2rs + s^2
  • 12(4r2−6rs)−(r−s)(2r+3s)\frac{1}{2}(4r^2 - 6rs) - (r - s)(2r + 3s)3s2−4rs3s^2 - 4rs
  • r2(6−s)−s(r2−5s)r^2(6 - s) - s(r^2 - 5s)6r2−2r2s+5s26r^2 - 2r^2s + 5s^2

Cevap

The correct matches pair the first expression with 6r2−5rs+6s26r^2 - 5rs + 6s^2, the second expression with 2r2−2rs+s22r^2 - 2rs + s^2, the third expression with 3s2−4rs3s^2 - 4rs, and the fourth expression with 6r2−2r2s+5s26r^2 - 2r^2s + 5s^2.
Each expression is correctly simplified by performing distribution first (carefully tracking negative signs and binomial expansion rules) and then combining terms that share the exact same variable powers.

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1
Simplify the expression 3r(2r−s)−2s(r−3s)3r(2r - s) - 2s(r - 3s).
6r2−5rs+6s26r^2 - 5rs + 6s^2
Distribute 3r3r to get 6r2−3rs6r^2 - 3rs. Then distribute −2s-2s to get −2rs+6s2-2rs + 6s^2 (noting that a negative times a negative is positive). Combine the like terms −3rs-3rs and −2rs-2rs to get −5rs-5rs.
2
Simplify the expression (2r−s)2−2(r2−rs)(2r - s)^2 - 2(r^2 - rs).
2r2−2rs+s22r^2 - 2rs + s^2
Square the binomial (2r−s)2(2r - s)^2 using the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to get 4r2−4rs+s24r^2 - 4rs + s^2. Distribute −2-2 to get −2r2+2rs-2r^2 + 2rs. Combine 4r2−2r24r^2 - 2r^2 to get 2r22r^2, and −4rs+2rs-4rs + 2rs to get −2rs-2rs.
3
Simplify the expression 12(4r2−6rs)−(r−s)(2r+3s)\frac{1}{2}(4r^2 - 6rs) - (r - s)(2r + 3s).
3s2−4rs3s^2 - 4rs
Distribute 12\frac{1}{2} to get 2r2−3rs2r^2 - 3rs. Expand the binomial product to get 2r2+3rs−2rs−3s2=2r2+rs−3s22r^2 + 3rs - 2rs - 3s^2 = 2r^2 + rs - 3s^2. Subtract this entire expression: (2r2−3rs)−(2r2+rs−3s2)=2r2−3rs−2r2−rs+3s2(2r^2 - 3rs) - (2r^2 + rs - 3s^2) = 2r^2 - 3rs - 2r^2 - rs + 3s^2. Combine like terms to get −4rs+3s2-4rs + 3s^2.
4
Simplify the expression r2(6−s)−s(r2−5s)r^2(6 - s) - s(r^2 - 5s).
6r2−2r2s+5s26r^2 - 2r^2s + 5s^2
Distribute r2r^2 to get 6r2−r2s6r^2 - r^2s. Distribute −s-s to get −sr2+5s2-sr^2 + 5s^2. Combine the like terms −r2s-r^2s and −sr2-sr^2 (since multiplication is commutative) to get −2r2s-2r^2s.

Anahtar Kavram

Simplifying expressions by applying the distributive property, expanding products of binomials, and combining like terms.
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Simplifying Expressions and Combining Like Terms

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